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Supersymmetric localization provides an efficient framework for computing partition functions of three-dimensional gauge theories on compact manifolds. For the case of lens spaces, these partition functions can be represented by using hyperbolic hypergeometric integrals. Comparing the resulting expressions for dual theories not only confirms infrared dualities but also gives rise to nontrivial identities among special functions. In this article, we analyze a set of such relations and demonstrate how they simplify, in appropriate limits, to ordinary hypergeometric identities. Our results highlight the role of dimensional reduction in connecting higher-dimensional dualities to lower-dimensional ones.

In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function ζ(s) at its point of symmetry. This hyperbolicity has been proved for degrees d ≤ 3. We obtain an asymptotic formula for the central derivatives ζ (2n)(1/2) that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all d ≤ 8. These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.

We consider directed polymers in random environment in the critical dimension d=2, focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on R2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise.

Andrews introduced the partition function $\\overline {C}_{k, i}(n)$ , called the singular overpartition function, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\\equiv \\pm i\\pmod {k}$ may be overlined. We prove that $\\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive integer k. We also prove that $\\overline {C}_{6, 2}(n)$ and $\\overline {C}_{12, 4}(n)$ are almost always divisible by $3^k$ . Using a result of Ono and Taguchi on nilpotency of Hecke operators, we find infinite families of congruences modulo arbitrary powers of $2$ satisfied by $\\overline {C}_{6, 2}(n)$ .

A bstract We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the N = 4 SU( N ) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the N = 2 ∗ theory with respect to the squashing parameter b and mass parameter m , evaluated at the values b = 1 and m = 0 that correspond to the N = 4 theory on a round sphere. At each order in the 1 /N expansion, these fourth derivatives are modular invariant functions of ( τ, τ ¯ ). We present evidence that at half-integer orders in 1 /N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1 /N , they are certain “generalized Eisenstein series” which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS 5 × S 5 .

A Maple package, which includes Radu’s algorithm and other algorithms of modular form tools, is provided. This enables us to prove or discover congruences for some restricted partition functions through building up Ramanujan-Kolberg identities, as well as perform many manipulations related to q-series and modular forms, such as dissection formulas and the Hecke operator. A pair of new partition congruences and their proof were found by this program as examples.

A bstract We study 3d N = 2 supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S 1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L ( p, q ) b with rational squashing parameter b 2 ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

A bstract We compute 1 /λ corrections to the four-point functions of half-BPS operators in SU( N ) N = 4 super-Yang-Mills theory at large N and large ’t Hooft coupling λ = g YM 2 N using two methods. Firstly, we relate integrals of these correlators to derivatives of the mass deformed S 4 free energy, which was computed at leading order in large N and to all orders in 1 /λ using supersymmetric localization. Secondly, we use AdS/CFT to relate these 1 /λ corrections to higher derivative corrections to supergravity for scattering amplitudes of Kaluza-Klein scalars in IIB string theory on AdS 5 × S 5 , which in the flat space limit are known from worldsheet calculations. These two methods match at the order corresponding to the tree level R 4 interaction in string theory, which provides a precise check of AdS/CFT beyond supergravity, and allow us to derive the holographic correlators to tree level D 4 R 4 order. Combined with constraints from [1], our results can be used to derive CFT data to one-loop D 4 R 4 order. Finally, we use AdS/CFT to fix these correlators in the limit where N is taken to be large while g YM is kept fixed. In this limit, we present a conjecture for the small mass limit of the S 4 partition function that includes all instanton corrections and is written in terms of the same Eisenstein series that appear in the study of string theory scattering amplitudes.

Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both1+1Dand higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In1+1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speedvB. We find that in1+1D, the “front” of the OTOC broadens diffusively, with a width scaling in time ast1/2. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization. Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales ast1/3in2+1Dand ast0.240in3+1D(exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in2+1D. We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical. However, when full spatial rotational symmetry is present in2+1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in1+1D, we map it to the partition function of an Ising-like statistical mechanics model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in1+1Dcircuits.

Graph partitioning is a theoretical subject with applications in many areas, principally: numerical analysis, programs mapping onto parallel architectures, image segmentation, VLSI design. During the last 40 years, the literature has strongly increased and big improvements have been made. This book brings together the knowledge accumulated during many years to extract both theoretical foundations of graph partitioning and its main applications.